Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function. (17th August 2010) - An An, Yulian Yulian; Luo Luo, Hua Hua
Abstract : An exact multiplicity result of positive solutions for the boundary value problems u ′′ + λ a ( t ) f ( u ) = 0, t ∈ ( 0, 1 ), u ′ ( 0 ) = 0, u ( 1 ) = 0 is achieved, where λ is a positive parameter. Here the function f : [ 0, ∞ ) → [ 0, ∞ ) is C 2 and satisfies f ( 0 ) = f ( s ) = 0, f ( u ) > 0 for u ∈ ( 0, s ) ∪ ( s, ∞ ) for some s ∈ ( 0, ∞ ) . Moreover, f is asymptotically linear and f ″ can change sign only once. The weight function a : [ 0, 1 ] → ( 0, ∞ ) is C 2 and satisfies a ′ ( t ) < 0, 3 ( a ′ ( t ) ) 2 < 2 a ( t ) a ′′ ( t ) for t ∈ [ 0, 1 ] . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for λ lying in various intervals in ℝ . Moreover, we indicate how to extend the result to the general case.